All Questions
9,056 questions
3
votes
0
answers
145
views
Formality of Sullivan Representatives
Suppose we have a map $f : \mathcal{A} \to \mathcal{B}$ between two formal, simply connected CDGAs, with induced map on cohomology $H(f) : H(\mathcal{A}) \to H(\mathcal{B})$. Further, suppose we have ...
- 285
7
votes
2
answers
750
views
Simplicial set construction of the classifying space
What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?:
Regard $G$ as a category with ...
- 275
61
votes
4
answers
10k
views
Hirzebruch's motivation of the Todd class
In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, ...
- 1,415
4
votes
0
answers
268
views
Is every tree a deformation retract of the disk?
I apologise if this question is not suitable for MathOverflow.
We define a graph here to mean a disjoint union of points and copies of $[0,1]$ quotiented so that the endpoints of any interval lie on a ...
- 91
6
votes
1
answer
387
views
Do all spaces doubly covered by $S^{2n}$ have the homeomorphism type of $\mathbb{P}^{2n}_{\mathbb{R}}$?
For reference, my motivation: It's of interest to classify free actions of groups on spheres of positive even dimension. Establishing such a classification up to homotopy is not too difficult: Every ...
- 233
2
votes
2
answers
382
views
Spaces homotopy dominated by $S^2 \times S^2\times S^2$
We say that a topological space $A$ is homotopy dominated by a topological space $X$ if there exist continuous maps $f:A\to X$ and $g:X\to A$ such that $g\circ f\simeq 1_A$.
Let $X$ be $S^2 \times S^2 ...
- 1,182
14
votes
2
answers
2k
views
What is the top cohomology group of a non-compact, non-orientable manifold?
Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$.
Question: Is the top singular cohomology group $H^n(M,\mathbb Z)$ zero?
This naïve question does not seem to ...
- 47.5k
7
votes
1
answer
2k
views
Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers
Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$.
If $f$ ...
- 10.7k
11
votes
2
answers
367
views
Spectrum $E$ with $H^\bullet(E,\mathbb{Z}/2)=\mathcal{A}//\mathcal{A}(n)$
Let $\mathcal{A}$ be the Steenrod Algebra and $\mathcal{A}(n)$ be the subalgebra generated by $Sq^1, Sq^{2}, Sq^{2^2},\ldots, Sq^{2^n}$.
It is known that
$H^*(H\mathbb{Z},\mathbb{Z}/2)=\mathcal{A}//\...
- 6,133
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 413
9
votes
1
answer
626
views
What motivated Thom to relate the cobordism groups with some homotopy groups?
I would like to know what motivated or led Thom to think that the (un)oriented cobordism groups would correspond with the homotopy groups of some structure (Thom spectum), or with the coefficient ...
- 601
5
votes
1
answer
461
views
Numerator in the zeta values at negative odd integers
The real J-homomorphism produces cyclic subgroups of size the denominator of $\zeta(1-2k)=-B_{2k}/k$ for $k>0$ in $\pi_{4k-1}S$ which completely account for first layer of the chromatic filtration ...
- 705
7
votes
1
answer
341
views
How can I detect the homology image of a unipotent group in the general linear group?
Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements.
Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
- 441
10
votes
1
answer
929
views
Is every additive cohomology operation stable?
To start, let's work with mod $p$ cohomology $H\mathbb F_p$ where $p$ is a prime. Consider the following three things:
The bigraded abelian group of all unstable cohomology operations, comprising all ...
- 64k
8
votes
1
answer
399
views
Which spectra arise from partially ordered commutative monoids?
Thomason showed how any connective spectrum arises from a symmetric monoidal category:
Robert W. Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), 78–...
- 22.3k
30
votes
5
answers
4k
views
The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
83
votes
0
answers
3k
views
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
- 22.3k
58
votes
10
answers
9k
views
de Rham cohomology and flat vector bundles
I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...
- 1,939
31
votes
2
answers
1k
views
Is Lie group cohomology determined by restriction to finite subgroups?
Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
- 413
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
2
votes
0
answers
70
views
Lifting paths along group quotients relative to a base
Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly ...
- 455
9
votes
0
answers
269
views
Colimits of symmetric groups
The infinite symmetric group $S_{\infty}$ of finitely supported permutations of $\mathbb{N}$ can be written as a colimit over the $S_n$'s with respect to the embedding $S_{n} \to S_{n+1}$ that maps $\...
- 7,637
0
votes
1
answer
240
views
Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book
Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...
- 6,023
7
votes
2
answers
213
views
Cofibrancy of a right module over an operad
If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module ...
- 5,849
7
votes
2
answers
927
views
Inclusion–exclusion principle for the compactly supported Euler characteristic
If $M$ and $N$ are sufficiently nice subspaces of some topological space $X$ then their Euler characteristics obey an inclusion-exclusion principle:
\begin{equation}
\chi(M) + \chi(N) = \chi(M\cup N) +...
- 133
22
votes
3
answers
820
views
Boardman's thesis or mimeographed notes
I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more ...
- 12.1k
9
votes
0
answers
182
views
What homology theory is calculated by unreduced cubical chains?
For a topological space $X$ and a subspace $A$, let $Q_n(X,A)$ be the group of singular cubical $n$-chains of $X$ relative to $A$ and let $D_n(X,A)$ be the subgroup of degenerate cubical chains. The ...
- 91
35
votes
5
answers
9k
views
Intuition behind Alexander duality
I was wondering if anyone could offer some intuition for why Alexander duality holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I ...
- 361
10
votes
1
answer
1k
views
Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles
I am trying to visualize the genus-two Riemann surface given by the curve
$$
y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}.
$$
We can regard this surface as a three-fold cover of the sphere with four ...
- 355
5
votes
2
answers
249
views
Patching up two trivial fibre bundles induces homology equivalence
I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...
- 585
18
votes
4
answers
3k
views
(Very) High dimensional manifolds
Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
12
votes
2
answers
767
views
Unique almost complex structure up to diffeomorphism
For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
user164740
43
votes
8
answers
5k
views
What part of the fundamental group is captured by the second homology group?
Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is ...
- 22.1k
2
votes
1
answer
264
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
- 7,527
18
votes
1
answer
783
views
Are there any "simple" monoids with intermediate growth?
The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
- 1,947
0
votes
2
answers
11k
views
Mathematics Roadmap [closed]
I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language.
My question is addressed to people who understand mathematics. I hope for ...
- 25
8
votes
3
answers
549
views
Contractible set in a manifold
Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(...
- 891
2
votes
0
answers
67
views
Künneth formula and continuity of the isomorphism
In the book Sheaf Theory, by Bredon (edition from 1997), Theorem 14.1, he writes a natural exact sequence, which, in some nice cases, leads to the Künneth formula. Do we have any reason to believe ...
1
vote
0
answers
128
views
Singular or cellular homology with $L^2$ coefficients
There are a few cases that $L^2$-homology (cohomology) that can be introduced. For example, for a manifold, it can be defined in the same way as de Rham cohomology using square integrable differential ...
- 161
7
votes
1
answer
662
views
Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory
Here is a collection of facts that all seem true, but together seem to give a nonsensical solution:
After $T(n)$-localization, all natural transformations $F \sim G$ between homogenous functors $F,G:...
- 5,849
48
votes
2
answers
8k
views
Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
- 1,098
14
votes
1
answer
862
views
Mapping torus of Klein bottle
This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...
- 4,081
2
votes
0
answers
354
views
Higher-order HKR theorems?
Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...
- 64k
-2
votes
1
answer
189
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 521
25
votes
2
answers
844
views
Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which ...
- 44.4k
9
votes
0
answers
405
views
What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
- 1,969
35
votes
9
answers
5k
views
Covering maps in real life that can be demonstrated to students
Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...
7
votes
1
answer
575
views
What is the closure of the Eilenberg-MacLane spectra under limits? under colimits?
Every bounded spectrum is in the closure of the Eilenberg MacLane spectra under finite co/limits. Thus every bounded below (resp. above) spectrum is in the closure of the EM spectra under limits (resp....
- 64k
2
votes
1
answer
164
views
Are there infinite number of 3-braids with trivial closure?
Not counting equivalent braids, are there finite or infinite numbers of 3-braids whose closures are trivial knot or links? If the answer is infinite, are there some patterns in those infinite numbers ...
- 73
3
votes
2
answers
309
views
Pullback of $w_1$ for 3-manifolds
Given closed $3$-manifolds $M$ and $N$
and an element $\alpha\in H^1(M;\mathbb{Z}_2)$,
when does there exist a map $f:M\to N$
such that $\alpha=f^*(w_1(N))$?
- 1,095